1. Consider the following linear regression model:

y_i = β₁ + β₂xi₂ + β₃xi₃ + ε_i = x_i′ β + ε_i

(a) Explain how the ordinary least squares estimator for β is determined and derive an expression for b.

(b) Which assumptions are needed to make b an unbiased estimator for β?

(c) Explain how a confidence interval for β₂ can be constructed. Which additional assump- tions are needed?

(d) Explain how one can test the hypothesis that β₃ = 1.

(e) Explain how one can test the hypothesis that β₂ + β₃ = 0. (f) Explain how one can test the hypothesis that β₂ = β₃ = 0.

(g) Which assumptions are needed to make b a consistent estimator for β?

(h) Suppose that x_i2 = 2 + 3x_i3. What will happen if you try to estimate the above model?

(i) Suppose that the model is estimated with x_i2 = 2x_i2 - 2 included rather than x_i2. How are the coefficients in this model related to those in the original model? And R2s?

(j) Suppose that x_i2 = x_i3 + u_i, where ui and x_i3 are uncorrelated. Suppose that the model is estimated with ui included rather than x_i2. How are the coefficients in this model related to those in the original model? And the R²s?

2. Using a sample of 545 full-time workers, a researcher is interested in the question as to whether women are systematically underpaid compared with men. First, she estimates the average hourly wages in the sample for men and women, which are $5.91 and $5.09 respectively.

(a) Do these numbers give an answer to the question of interest? Why not?

The researcher also runs a simple regression of an individual's wage on a male dummy, equal to 1 for males and 0 for females. This gives the results reported in Table 1.

(b) How can you interpret the coefficient estimate of 0.82? How do you interpret the esti- mated intercept of 5.09?

(c) How do you interpret the R2 of 0.26?

(d) Explain the relationship between the coefficient estimates in the table and the average wage rates of males and females.

(e) A student is unhappy with this model as ‘a female dummy is omitted from the model'. Comment upon this criticism.

(f) Construct a 95% confidence interval for the average wage differential between males and females in the population.

Subsequently, the above ‘model' is extended to include differences in age and education by including the variables age (age in years) and educ (education level, from 1 to 5). Simultane- ously, the dependent variable is adjusted to be the natural logarithm of the hourly wage rate. The results are reported in Table 2.

(g) How do you interpret the coefficients of 0.13 for the male dummy and 0.09 for age?

(h) A student is unhappy with this model as ‘the effect of education is rather restrictive'. Can you explain this criticism? How could the model be extended or changed to meet the above criticism? How can you test whether the extension has been useful?

The researcher re-estimates the above model including age² as an additional regressor. The t- value on this new variable becomes -1.14, while R²_a increases to 0.683.

(i) Could you give a reason why the inclusion of age2 might be appropriate?

(j) Would you retain this new variable given the R²_a measure? Would you retain age2 given its t-value? Explain this apparent conflict in conclusions.

Table 1: Hourly wages explained from gender: OLS results

N = 545 s = 2.17 R²a = 0.26

Table 2: Log hourly wages explained from gender, age and education level: OLS results

N = 545 s = 0.24 R²_a = 0.682

3. The most important assumption for small sample OLS regression is the exogeneity of explana- tory variables. Consider the following regression equation based on a random sample,

y_i = β₀ + β₁x_1i + β₂x_2i + ε_i

In order for OLS to yield unbiased estimates, we need to assume that both x1 and x2 are exogenous to the error term ε, i.e.,

E(ε_i|x_1i, x_2i) = E(ε_i) = 0 (1)

Sometimes our main interest is in the marginal effect of x₁ on y, with x₂ only serving as a control variable. However, if x₂ is not exogenous to the error term, the estimate of β₂ will be biased. Moreover, the estimates of other coefficients, especially β₁, will also be contaminated, which invalidates the entire research design.

In practice, the control variable x₂ is very likely to be endogenous, i.e., assumption (1) does not hold. But if we are willing to impose the following assumption instead,

E(εi|x_1i, x_2i) = E(ε_i|x_2i) (2)
we can still guarantee that the estimate of β₁ is unbiased. This assumption is named "condi- tional mean independence". It implies that, once we control for x2, we can safely regard x₁ as exogenous. For simplicity, let us assume further that

E(ε_i|x_2i) = γ₀ + γ₁x_2i
(a) Why is the conditional mean independence assumption (2) sufficient for the unbiasedness of the OLS estimate of β₁?

(b) Does the conditional mean independence assumption (2) also guarantee unbiased esti- mates of β₀ and β₂?

4. Consider the following simple regression model

yi = β₀ + β₁x_i + ε_i, i = 1, . . . , n

{y_i, x_i} are randomly sampled from the population. Assume that E(x_iε_i) = 0, Var(x_i) = σ_x² > 0, Var(ε_i) σ²_ε > 0.

In practice, a student mistakenly treated x as the dependent variable, and y as the independent variable and obtained the OLS estimate b of the coefficient on y. When she realized the mistake, she used β^{^}₁ = b^-1 as an estimate of β₁.

(a) Show that β^{^}1 is not a consistent estimator of β₁.

(b) Show the direction of bias in β^₁.

5. According to the great economist John Maynard Keynes, consumption per capita in the economy is a linear function of per capita income, i.e.,

C_t = β₀ + β₁Yt + εt (3)

t = 1, . . . , T denotes period of measure. ε assembles all the non-income factors that have an impact on consumption. β₁ is called marginal propensity of consumption, 0 < β₁ < 1. This function implies a causal relationship between consumption and income: by how much will consumption increase if income increases by one unit? Meanwhile, we know from the national income identity that income is comprised of consumption and investment (assume the absence of government), i.e.,

Y_t = C_t + I_t

where It denotes per capita investment. Assume investment is exogenous, i.e.,

E(I_tε_t) = 0

Give a critical comment on the research design that tries to estimate the marginal propensity of consumption by running an OLS regression with respect to (3). In particular, will this sort of research over-estimate or under-estimate the true marginal propensity of consumption?